Carroll was sent off after it was deemed he had struck Swansea defender Chico Flores and the Hammers, upset that the FA did not overturn the decision, took the matter to an independent arbitral tribunal on Friday seeking what manager Sam Allardyce claimed would be "justice".
However, the independent panel, at which the FA was also represented, found there was "no serious issue to be tried" and as such rejected West Ham's submissions, meaning Carroll must serve out his punishment.
The first game of that suspension was the 2-0 victory over Aston Villa on Saturday and Nolan, who scored both goals at Villa Park, still believes the FA was wrong not to overturn the initial decision.
"We have nothing against the FA, they've just got to get it right," he told several national newspapers.
"When people get things wrong we expect them to hold up their hands and say, 'Yeah, I got it wrong' and move on, like we all have to do.
"We have to hold our hands up as footballers when we get it wrong and take the stick that comes with it. I think they had a chance to redeem themselves and they haven't taken it.
"We were disappointed with the outcome. We thought it was an injustice. Andy is also frustrated, he's a good mate of mine and I felt really sorry for him.
"We've got to forget about Andy for the next couple of weeks now. He has to work hard and get himself ready because if we keep winning games it's going to be hard for him to get back into the team."